Remark 3.2 For any \(i\in I\) , Lusztig (Section 1.2.13 in [1]}) introduced a derivation \({_ir}:\mathbf {U}^-\rightarrow \mathbf {U}^-\)  such that
\({_ir}^{m}(xy)=\sum _{t=0}^{m}v^{(\nu -ti,(m-t)i)+t(m-t)}\frac{[m]_{v}!}{[t]_{v}![m-t]_{v}!}{_ir}^{t}(x){_ir}^{m-t}(y)\) 
